Statistical Inference for Interval Identified Parameters

This paper analyzes the construction of confidence intervals for a parameter theta(0) that is "interval identified," that is, the sampling process only reveals upper and lower bounds on theta(0) even in the limit. Analysis of inference for such parameters requires one to reconsider some fundamental issues. To begin, it is not clear which object - the parameter or the set of parameter values characterized by the bounds - should be asymptotically covered by a confidence region. Next, some straightforwardly constructed confidence intervals encounter problems because sampling distributions of relevant quantities can change discontinuously as parameter values change, leading to problems that are familiar from the pre-testing and model selection literatures. I carry out the relevant analyses for the simple model under consideration, but also emphasize the generality of problems encountered and connect developments to general themes in the rapidly developing literature on inference under partial identification. Results are illustrated with an application to the Survey of Economic Expectations.